# Trigonometry Formulas

In Trigonometry, different types of problems can be solved using trigonometry formulas. These problems may include trigonometric ratios (sin, cos, tan, sec, cosec and tan), Pythagorean identities, product identities, etc. Some formulas including the sign of ratios in different quadrants, involving co-function identities (shifting angles), sum & difference identities, double angle identities, half-angle identities, etc. are also given in brief here.

Learning and memorizing these formulas will help the students of Class 10, 11, and 12 to score good marks in this concept. They can find the trigonometry table along with inverse trigonometry formulas to solve the problems based on them.

Trigonometry is a branch of mathematics that deals with triangles. Trigonometry is also known as the study of relationships between lengths and angles of triangles.

There is an enormous number of uses of trigonometry and its formulae. For example, the technique of triangulation is used in Geography to measure the distance between landmarks; in Astronomy, to measure the distance to nearby stars and also in satellite navigation systems.

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## Trigonometry Formulas List

When we learn about trigonometric formulas, we consider it for right-angled triangles only. In a right-angled triangle, we have 3 sides namely – Hypotenuse, Opposite side (Perpendicular), and Adjacent side (Height). The longest side is known as the hypotenuse, the side opposite to the angle is perpendicular and the side where both hypotenuse and opposite side rests is the adjacent side.

Here is the list of formulas for trigonometry.

### Basic Formulas

There are basically 6 ratios used for finding the elements in Trigonometry. They are called trigonometric functions. The six trigonometric functions are sine, cosine, secant, co-secant, tangent and co-tangent.

By using a right-angled triangle as a reference, the trigonometric functions or identities are derived:

• sin θ = Opposite Side/Hypotenuse
• cos θ = Adjacent Side/Hypotenuse
• tan θ = Opposite Side/Adjacent Side
• sec θ = Hypotenuse/Adjacent Side
• cosec θ = Hypotenuse/Opposite Side
• cot θ = Adjacent Side/Opposite Side

### Reciprocal Identities

The Reciprocal Identities are given as:

• cosec θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ
• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ

All these are taken from a right angled triangle. With the height and base side of the right triangle given, we can find out the sine, cosine, tangent, secant, cosecant, and cotangent values using trigonometric formulas. The reciprocal trigonometric identities are also derived by using the trigonometric functions.

### Trigonometry Table

Below is the table for trigonometry formulas for angles that are commonly used for solving problems.

 Angles (In Degrees) 0° 30° 45° 60° 90° 180° 270° 360° Angles (In Radians) 0° π/6 π/4 π/3 π/2 π 3π/2 2π sin 0 1/2 1/√2 √3/2 1 0 -1 0 cos 1 √3/2 1/√2 1/2 0 -1 0 1 tan 0 1/√3 1 √3 ∞ 0 ∞ 0 cot ∞ √3 1 1/√3 0 ∞ 0 ∞ csc ∞ 2 √2 2/√3 1 ∞ -1 ∞ sec 1 2/√3 √2 2 ∞ -1 ∞ 1

These formulas are used to shift the angles by π/2, π, 2π, etc. They are also called co-function identities.

• sin (π/2 – A) = cos A & cos (π/2 – A) = sin A
• sin (π/2 + A) = cos A & cos (π/2 + A) = – sin A
• sin (3π/2 – A)  = – cos A & cos (3π/2 – A)  = – sin A
• sin (3π/2 + A) = – cos A & cos (3π/2 + A) = sin A
• sin (π – A) = sin A &  cos (π – A) = – cos A
• sin (π + A) = – sin A & cos (π + A) = – cos A
• sin (2π – A) = – sin A & cos (2π – A) = cos A
• sin (2π + A) = sin A & cos (2π + A) = cos A

All trigonometric identities are cyclic in nature. They repeat themselves after this periodicity constant. This periodicity constant is different for different trigonometric identities. tan 45° = tan 225° but this is true for cos 45° and cos 225°. Refer to the above trigonometry table to verify the values.

### Co-function Identities (in Degrees)

The co-function or periodic identities can also be represented in degrees as:

• sin(90°−x) = cos x
• cos(90°−x) = sin x
• tan(90°−x) = cot x
• cot(90°−x) = tan x
• sec(90°−x) = csc x
• csc(90°−x) = sec x

### Sum & Difference Identities

• sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
• cos(x+y) = cos(x)cos(y)–sin(x)sin(y)
• tan(x+y) = (tan x + tan y)/ (1−tan x •tan y)
• sin(x–y) = sin(x)cos(y)–cos(x)sin(y)
• cos(x–y) = cos(x)cos(y) + sin(x)sin(y)
• tan(x−y) = (tan x–tan y)/ (1+tan x • tan y)

### Double Angle Identities

• sin(2x) = 2sin(x) • cos(x) = [2tan x/(1+tan2 x)]
• cos(2x) = cos2(x)–sin2(x) = [(1-tan2 x)/(1+tan2 x)]
• cos(2x) = 2cos2(x)−1 = 1–2sin2(x)
• tan(2x) = [2tan(x)]/ [1−tan2(x)]
• sec (2x) = secx/(2-sec2 x)
• csc (2x) = (sec x. csc x)/2

### Triple Angle Identities

• Sin 3x = 3sin x – 4sin3x
• Cos 3x = 4cos3x-3cos x
• Tan 3x = [3tanx-tan3x]/[1-3tan2x]

### Half Angle Identities

• $\sin\frac{x}{2}=\pm \sqrt{\frac{1-\cos\: x}{2}}$
• $\cos\frac{x}{2}=\pm \sqrt{\frac{1+\cos\: x}{2}}$
• $\tan(\frac{x}{2}) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}$

Also, $\tan(\frac{x}{2}) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}\\ \\ \\ =\sqrt{\frac{(1-\cos(x))(1-\cos(x))}{(1+\cos(x))(1-\cos(x))}}\\ \\ \\ =\sqrt{\frac{(1-\cos(x))^{2}}{1-\cos^{2}(x)}}\\ \\ \\ =\sqrt{\frac{(1-\cos(x))^{2}}{\sin^{2}(x)}}\\ \\ \\ =\frac{1-\cos(x)}{\sin(x)}$ So, $\tan(\frac{x}{2}) =\frac{1-\cos(x)}{\sin(x)}$

### Product identities

• $\sin\: x\cdot \cos\:y=\frac{\sin(x+y)+\sin(x-y)}{2}$
• $\cos\: x\cdot \cos\:y=\frac{\cos(x+y)+\cos(x-y)}{2}$
• $\sin\: x\cdot \sin\:y=\frac{\cos(x-y)-\cos(x+y)}{2}$

### Sum to Product Identities

• $\sin\: x+\sin\: y=2\sin\frac{x+y}{2}\cos\frac{x-y}{2}$
• $\sin\: x-\sin\: y=2\cos\frac{x+y}{2}\sin\frac{x-y}{2}$
• $\cos\: x+\cos\: y=2\cos\frac{x+y}{2}\cos\frac{x-y}{2}$
• $\cos\: x-\cos\: y=-2\sin\frac{x+y}{2}\sin\frac{x-y}{2}$

### Inverse Trigonometry Formulas

• sin-1 (–x) = – sin-1 x
• cos-1 (–x) = π – sin-1 x
• tan-1 (–x) = – tan-1 x
• cosec-1 (–x) = – cosec-1 x
• sec-1 (–x) = π – sec-1 x
• cot-1 (–x) = π – cot-1 x

## Trigonometry Formulas Major systems

All trigonometric formulas are divided into two major systems:

• Trigonometric Identities
• Trigonometric Ratios

Trigonometric Identities are formulas that involve Trigonometric functions. These identities are true for all values of the variables. Trigonometric Ratio is known for the relationship between the measurement of the angles and the length of the sides of the right triangle.

Here we provide a list of all Trigonometry formula for the students. These formulas are helpful for the students in solving problems based on these formulas or any trigonometric application. Along with these, trigonometric identities help us to derive the trigonometric formulas, if they will appear in the examination.

We also provide the basic trigonometric table pdf that gives the relation of all trigonometric functions along with their standard values. These trigonometric formulae are helpful in determining the domain, range, and value of a compound trigonometric function. Students can refer to the formulas provided below or can also download the trigonometric formulas pdf that is provided above.

## Frequently Asked Questions – FAQs

### What are the basic trigonometric ratios?

Sine, Cosine, Tangent, Cotangent, Secant and Cosecant.

### What are formulas for trigonometry ratios?

Sin A = Perpendicular/Hypotenuse
Cos A = Base/Hypotenuse
Tan A = Perpendicular/Base

### What are the three main functions in trigonometry?

Sin, Cos and Tan are three main functions in trigonometry.

### What are the fundamental trigonometry identities?

The three fundamental identities are:
1. sin^2 A + cos^2 A = 1
2. 1+tan^2 A = sec^2 A
3. 1+cot^2 A = csc^2 A

### Trigonometry formulas are applicable to which triangle?

Right-angled triangle

Trigonometric equations chapter PDF

1. lavanya
3. pawar Avinash

Very usefully too others students

4. The formulas mentioned here were very handy and my kid learns in a quick way.

1. byjus developer

Thank you for the response

5. keshav

please u can give all formula of trigonometry chapter

6. Ehsaas Gupta

where are the triple term formulae?

7. Please tell me that only these formulas are sufficient for any college entrance exam

Proper Content for quick learning and Revision

10. Animesh scis

11. Arnav Minotra

Thanks to byjus

12. ROUSHAN CHOUBEY